How do you find the contribution of an increase of each term within a cube root?
Given that: $$result = \sqrt[3]{a*b*c}$$
Looking at how the result changes from say t0 to t1:
$$result_0 = \sqrt[3]{a_0*b_0*c_0}$$
$$result_1 = \sqrt[3]{a_1*b_1*c_1}$$
I want to know how the change in each term affects the change in the result.
$(a_1-a_0)$ contributed ___% to the change $(result_1-result_0)$
$(b_1-b_0)$ contributed ___% to the change $(result_1-result_0)$
$(c_1-c_0)$ contributed ___% to the change $(result_1-result_0)$
e.g. Let’s say $(a_1-a_0) > (b_1-b_0) > (c_1-c_0)$
Obviously then the change in a $(a_1-a_0)$ contributes a larger percentage than the change in b $(b_1-b_0)$ and in c $(c_1-c_0)$ to the change in the result characterized by $(result_1-result_0)$
Is it just $\frac{(a_1-a_0)}{(a_1-a_0) + (b_1-b_0) + (c_1-c_0)}$?
Also: what if, say, $(c_1-c_0)$ is negative? How does that affect the calculation?
Thanks very much!
EDIT - To Clarify My Question
Let's JUST SAY, between two periods of time:
a is running ability, b is swimming ability, c is climbing ability
result is overall athletic ability
$$\sqrt[3]{0.408*0.405*0.754} = 0.499= athleticability_0$$
$$\sqrt[3]{0.739*0.631*0.861} = 0.738 = athleticability_1$$
$$(athleticability_1-athleticability_0) = 0.239$$ $$(run_1-run_0) = 0.331$$ $$(swim_1-swim_0) = 0.226$$ $$(climb_1-climb_0) = 0.107$$ So my overall athletic ability increased by 0.239 because running ability increased by 0.331, swimming ability increased by 0.226, and climbing ability increased by 0.107. Does that imply:
$$\frac{(0.331)}{(0.331) + (0.226) + (0.107)} = 0.498 $$
Improvement to running ability accounted for 49.8% of that athletic ability improvement of 0.239?
(And so, using same method of calculation, increases in swimming ability accounted for 34.0% and increase in climbing ability accounted for 16.1%?)
The problem is more complex. The increment of the result depends also from the initial values, $a,b,c$
For instance, consider $\sqrt[3]{8\cdot 20\cdot 50}=20$
Increment the numbers by $2,3,4$ to get $\sqrt[3]{10\cdot 23\cdot 54}\approx 23.1583$
Now consider $\sqrt[3]{2\cdot 10\cdot 12}\approx 6.21447$
and apply the same increments. We get $\sqrt[3]{4\cdot 13\cdot 16}\approx 9.40534$
As you can see percentages are very different.
If you want to understand further in a easier way, start with only one variable, like $\sqrt[3]{x}$
Hope this helps